求解带PDE约束最优控制问题的数值方法研究

Optimal control models with PDE constraints have a wide range of applications in the fields of defense, meteorology, automotive engineering, and some problems related to the economics and social development. For the optimal control problem constrained by an elliptic equation, the corresponding discrete problem is a large-scale coupled system, which cannot be solved efficiently. If the constraint is a parabolic equation or an elliptic equation with random coefficients, the dimension of the finite dimensional problem will have polynomial growth with respect to the degree of freedom in the spatial detection, which makes it almost impossible to be solved directly. Our project studies the above problems, and proposes efficient algorithms corresponding to the properties of these problems. For the deterministic case, we shall apply the ADMM and the adjusted full Jacobi decomposition to solve the resulting discretized systems based on the variable separable structure, which reduces the matrix scale and improves the computational efficiency. For the random case, we shall transform the problem into a deterministic one by the multi-modes expansion and preprocessing technique, which overcomes the inherent drawbacks of the Monte Carlo method, such as the slow convergence and high storage cost. We will also present the convergence estimates for the proposed methods. The numerical simulations will be provided to verify the efficiency of our algorithms. This project shall develop numerical methods for partial differential equations, optimization and related interdisciplinary research, provide theoretical and technical support for many practical problems, and therefore it has theoretical significance as well as practical value.

带PDE约束的最优控制模型在国防、气象、以及汽车工程等与国民经济和社会发展密切相关的领域均有广泛应用。当约束为椭圆方程时，离散后对应一个大规模耦合系统，很难求解。当约束为抛物方程或含随机系数的椭圆方程时，对应离散问题的规模随节点数量呈多项式增长，直接求解几乎不可能。本项目主要研究以上几类问题，并针对问题的特点设计相应的高效算法。对确定问题，基于带求变量的可分结构，拟用ADMM和带矫正的全分裂Jacobi迭代法求解，缩小矩阵规模，提高计算效率。对随机问题，拟用多重模式展开及预处理技术将其化为确定问题求解，从本质上解决Monte Carlo方法收敛速度慢、存储要求高的缺欠。本项目还将对所提算法进行全局收敛性分析和数值模拟，以期达到理论创新和方法创新的统一。本项目的执行不仅可以促进PDE数值解及优化等交叉学科研究的发展，还能为许多实际问题提供理论保障和技术借鉴，具有理论意义和应用价值。

偏微分方程(PDE)约束最优控制模型在实际问题中有广泛应用。研究带PDE约束的最优控制问题，提出高效的数值算法是国民经济和社会发展的迫切需求，也是计算数学领域中的一个重要研究课题。本项目重点研究三类有代表性PDE最优控制问题的数值解法，完成了以下工作：1.关于Laplace方程最优控制问题，根据其状态和控制可分的特点，设计了求解该问题的乘子交替方向法(ADMM)，在AAMM上发表论文1篇。2.着眼于应用领域，将求解Laplace方程最优控制问题的算法应用到Maxwell方程最优问题中，在CICP上发表论文1篇。3.关于系数含随机的椭圆方程最优控制问题，构造出了三种合理的数值格式，并通过比较，找到了高效求解方案，在ERA上发表论文1篇，主要贡献在于利用多重模式展开，将随机问题转化成定问题，有效降低了存储和计算压力。4.项目组设计了针对抛物型最优控制问题的并行算法，已完成理论分析和初步数值测试，正在进行大规模并行模拟。5.项目进行过程中，项目组成员关注到了原始对偶积极集算法、牛顿迭代法、弱有限元，以及Nedelec元离散等技术，并将它们用于求解期权定价和流体等相关问题，共在JCP和CNSNS等杂志上发表论文4篇。目前为止，项目组基本按计划完成了所有工作，正在尝试将所得结果应用到更具实际意义的非线性PDE约束最优控制问题的求解中。